That would be Euclidean geometryAlso, is there a special term for the geometry typically taught in high school?
Yes, topology is also a kind of geometry. To see where things fit in: http://en.wikipedia.org/wiki/Erlangen_programAnd how does topology fit in here? It seems that topology is a form of geometry as well.
I believe the usual title is plane geometry, though admittedly it is pretty "plain"!Geometry taught in high school is "plain geometry" and a follow-on course (or a section late in the plain geometry course) would be "spherical geometry"
No. Algebraic geometry is a completely different course- and considerably more advanced.Analytic and algebraic geometry are the same thing (or at least that's how the words were used 50+ years ago when I was in high school).
Those are high school topics. Differential geometry and topology are much more advanced.
You really should learn how to use Google. It will answer such questions for you pretty readily. For example, Google "what is differential geometry"
It helped a bit.swampwiz: did my explanation help at all?
is this related to the Erlanger program http://en.wikipedia.org/wiki/Erlangen_program ?trying again to be more elementary: the geometries you mention differ roughly in how many derivatives are required for the functions used in them.
in topology no derivatives are required, just continuity.
in differential geometry at least two derivatives are needed and often infinitely many are used.
in analytic geometry not only are infinitely many derivatives required, but it is also required that the taylor series they define actually represents the function locally everywhere, i.e. an analytic function must be determined just by the values of all its derivatives at one point.
in algebraic geometry, not only are infinitely many derivatives required, but it also required that for some finite n >0, all derivatives of order > n are identically zero.